Calculation of the bloch effective Hamiltonian as an expectation value

Chemical Physics 127 (1988) 9-16 North-Holland, Amsterdam

CALCULATION OF THE BLOCH EFFECTIVE AS AN EXPECTATION VALUE Florent

Xavier GADEA

and Jean-Louis

Laboratoire de Physique Quantique,

HAMILTONIAN

HEULLY

Universite Paul-Sabatier,

118, route de Narbonne, 31062 Toulouse Cedex, France

Received 6 May 1988; in final form 1 August 1988

Using the symmetry between the model space and the target space, a new effective Hamiltonian is derived. It involves the usual wave operator D as determined order by order from the generalized Bloch equation and gives at convergence the usual Bloch effective Hamiltonian. Numerical tests on N, CI matrices are performed and compared to the usual results. The convergence series is improved and gives upper bounds for the energies. Its size consistency is tested on a model system.

1. Introduction The theory of effective Hamiltonian represents a major tool in many fields of quantum physics. It makes possible the derivation of the exact energies and wavefunctions of severals states from a single secular equation. In this theory the concepts of model space and wave operator play a crucial role. From the wave operator, defined on the model space, the effective Hamiltonian is built. In atomic and molecular physics, the most traditional and fundamental effective Hamiltonian is the Bloch Hamiltonian [l-4]. However, with the same wave operator another effective Hamiltonian can be built [5,6]. This Hamiltonian will be presented in this paper where we shall show that it has two main properties. First. in a perturbative derivation, it gives more reliable information than the Bloch Hamiltonian (the two being identical at convergence). Second, it guarantees upper bounds for all states of the model space.

2. Perturbation theory and effective Hamiltonian The aim of perturbation H$‘=

theory is to yield an approximate

solution

of the Schrodinger

E’#,

of a physical system with an arbitrary degree of accuracy. For that purpose, two parts, an approximate or model Hamiltonian H, and a perturbation V H=

Ho+

V,

HoRi=

the Hamiltonian

P,=CIR~)(R~I,

(i=1,2

qi are linear

the model space

Q,=i-PO.

If the model space is of dimension d, one can show that a number Hamiltonian have their major components in the model space, P&=#,

is split into

E;R’.

The known eigenfunctions of Ho are used to partition the Hilbert space into two parts: (S,) and its orthogonal complement (St) characterized by the projection operators

where

equation,

,...,

d),

combinations

of the 1R) vectors.

0301-0104/88/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

d of the eigenfunctions

of the full

F. X. Gadea, J.-L. Heully / Calculation of the Bloch effectioe Hamiltonian

10

The wave operator Q$‘,=+

52 transforms

(i=1,2

This wave operator

,...,

This equation holds expansion. The energies the model space,

into the exact ones,

d).

is a solution

[ii?, H”]PO = (viz-

the model functions

of the so called generalized

Bloch equation

[7]

(1)

mm)P,.

for all the d states of the model space and it is suitable for a perturbative of interest are obtained by diagonalizing the Bloch effective Hamiltonian [I,31 in

HB140) = E I #o>>HB

= PoHDPo = PoHoPo + P,VOP,.

This operator reproduces all the d eigenvalues of the full Hamiltonian corresponding to the $ states. This implies that if we know the wave.operator (whith sufficient accuracy), we can obtain the complete level structure corresponding to one or several configurations by solving a single secular equation of finite dimensionality. Eq. (1) (the generalized Bloch equation) allows a perturbative or iterative determination of the wave operator. In this section we shall see how it is possible to use this wave operator to get a more accurate effective Hamiltonian (EH) than the usual one [5,6]. The starting point is a general view of the subspaces involved in EH theory and their correspondances. This theory uses two partitions of the Hilbert space: the first one is the (PO, Q,) representation. defined by the model space (S,) and its orthogonal complement (SOL), the second one is the (P, Q) representation defined by the target space (S, the d exact eigenfunctions of H) and its orthogonal complement. The first representation (PO, Q,) is used explicitly since it is the only one known and all the vectors and operators will be determined in this representation. The second one is used implicitly since the target space represents the quantity we are looking for. Assuming that S and S, have the same dimension and are non-orthogonal - that means that if h is a basis of S,, Pb (its projection onto S) is a basis of S and vice versa if c is a basis of S, P,c is a basis of S, the projectors PO and P ensure a one-to-one correspondance between S and S,. Using the linearity of D it is easy to show that L’ acting on the ) R) vectors gives a basis of S, namely the biorthogonal basis of the 1PR) basis, noted ]m). These correspondances are illustrated in fig. 1. We can now work in the target space, and determine the contravariant matrix of H in the JPR) basis. By contravariant matrix we mean the contravariants of the I HPR) vectors in the (m) basis of S. This matrix is identical to the matrix of the Bloch effective Hamiltonian in the 1R) basis of S,, (PR,lHIPR,)=(RjIPoHf21R,)

(i=1,2

,... d; j=l,2

,...,

d).

(2)

Fig. 1. Illustration of the correspondances between the subspaces SO (model space) and S (target space).

F. X. Gadea, J.-L. Heully / Calcularion of the Bloch effective Hamiltonian

(i=

11

The matrix of &? in the (PO, Q,) representation gives directly the 1PR) vectors since Q 1R j) = 1PR,) 1,2 ,**., d), and the 1PR) vectors are simply determined as the biorthogonal set of the I OR) vectors,

i.e.

(PR,IHIPR,) =(S~(S~+~~-‘R~IH~IR~) (i=1,2

,...,

d; j=l,

2 ,...)

=(R,I(~~+~)-IQ+H~~IR,)

d).

So the left-hand side of eq. (2) yields a new effective Hamiltonian when s2 is converged. Let us call this effective Hamiltonian HBG,

which is identical

to the Bloch EH

HBG = (0+9)-‘L?+HS2; which can be illustrated

by the following

while the usual Bloch effective

matrix

Hamiltonian

representation,

is given by

It is clear that the determination of HBG mvolves . the whole matrix of H in the (PO, Qo) representation instead of the P,,H matrix for HB. Therefore given a 52, the determination of HBG is more time-consuming than the determination of HB, but since the IPR) vectors form a basis of the target space, the matrix (R; I HBG I R,) ( = (PR, I H I-) (i, j = 1, d)) is an expectation value of the Hamiltonian. This point is

Table 1 Comparison of convergence between H BG ( left) and HB significant figures are given E2

n

El

2 3 4 5 6 7 8 9 10 18

0.819 0.8183 0.8182 0.81819 0.81819 0.818187 0.818186 0.8181856 0.8181854

0.81 0.82 0.818 0.818 0.8181 0.8182 0.8182 0.81819 0.81818 0.8181853 0.8181853

(right) for the 20x20 test matrix defined in the text. At each order n

E3 0.89 0.891 0.990 0.890 0.8900 0.8900 0.89004 0.89004 0.8900036 0.8900354 0.89003537

0.890 0.890 0.8905 0.89003 0.89003 0.89CKI35 0.890034 0.8900353 0.8900353

E4 0.99 0.99 0.090 0.991 0.991 0.991 0.9909 0.9909 0.99093 0.9909395 0.9909394

0.992 0.991 0.9910 0.99097 0.99095 0.990941 0.9909399 0.9909395 0.9909394

E5 1.089 1.090 1.0892 1.0893 1.0893 1.0893 1.08925 1.08925 1.08925 1.0892476 1.0892475

1.090 1.0894 1.0893 1.08925 1.08925 1.089248 1.089247 1.0892476 1.0892475

1.162 1.164 1.162 1.163 1.1628 1.1629 1.16287 1.16287 1.16287 1.1628693 1.1628692

1.164 1.164 1.1629 1.16288 1.16287 1.16287 1.1628697 1.1628694 1.1628692

F. X. Gadea, J.-L. Heully / Calculation of the Bloch effective Hamiltonian

12

very clear for a one-dimensional model space. In this case there is only one normalized space ( 4,) and it is easy to get it from any vector of this one-dimensional subspace,

IPR,)= I QRJ> I PR,)=l~)=(nR,I~R,)-‘I~R,). since

1QR,)

belongs

vector in the target

1~,)=(~;2R,lS2R,)-“*I~‘R,),

to the target space,

(~,I~BGI~~)=(~~~l~I~)=(~~,/(~,l~+~l~,)I~l~~,)

This relation which should be true at convergence is also valid, in the sense that it produces an expectation value, at any level of approximation for 9. This new EH will thus yield upper bounds to the energy as variational calculations do In the case where there are several states in the model space the situation is not so clear. However one can expect that the variational principle is still valid in this case. In all the calculations we have performed, even in the case of strong divergences, all the roots of HRG were above the exact energies. With the usual Bloch EH, when the wave operator includes the perturbation up to order n, the energy is not so evident. Nevertheless as we use includes it up to the order (n + 1). With HBG, the classification the same D we will call “result at order (n + l)“, the HB” calculated from Q = C:_,@‘).

3. Calculation and discussion The first comparative test between the Bloch effective Hamiltonian real symmetric matrix with five vectors in the model space and fifteen elements Hi, are defined by h,,=l

(l
h,_,,,=O.l

h,,=2+1/10(i-6), h,, = i - 8,

hi--5,,=0.1 h,_lo., =O.l,

and the other form was made on a in the orthogonal space. The matrix

(2
h,_,,,=h,p,,,=0.5

(ll
is The results in table 1 illustrate the improved convergence of HRG. The origin of this improvement clearly related to the greater powers of V involved in H BG compared to H”. If Q involves V up to n, HBG involves it to 2n + 1 for the numerator and 2n for the denominator, whereas HB involves V only to n + 1. From table 1 we see that in order to get the same convergence, one needs with HBG a perturbation order which is half of the corresponding H B order. Fig. 2 illustrates the convergence pattern of the two effective Hamiltonians. One can see that for all the five roots, HBG gives upper bounds. This implies that some kind of variational principle is respected. In a second test we used a more physical case. We choose the N, molecule in a minimal basis and for non-perturbed Hamiltonian the unrestricted Hartree-Fock. Such a Hamiltonian gives wavefunctions which separate correctly and, consequently, correlated methods employing such reference functions generally yield good dissociation energies. However, UHF wavefunctions can have a large spin contamination especially in the intermediate bond-breaking region and as a result, spurious barriers are found even in highly correlated calculations. Furthermore it has been observed that perturbation theory based on UHF functions has a very slow convergence behaviour [8%10]. We are thus imposing to a very difficult test the HB and H BG effective Hamiltonians.

F. X. Gadea, J.-L. Heully / Calculation of the Bloch effective Hamiltonian

I

Al

J Fig. 2. Convergence

of the perturbation

13

.b

1 series for the five roots of the model space, the usual bloch other line giving the HBG.

HB is given by the thin line. The

Table 2 shows the correlation energy for the ground state of N, at different bond distances. Results are given at third order for HBG where the computational effort is intermediate between third- and fourth-order HB which is commonly used nowadays. Though fourty-order HB is more time-consuming than third order HBG, the latter g’v 1 es better results. After the point of RHF instability (R = 2.2) the wave operator converges very slowly and therefore, only small but significant differences are obtained between HBG and HB. In order to test the behavior of HBG we fixed a particular intermediate distance (r = 2 au) and chose an Epstein-Nesbet partitioning of the Hamiltonian [ll]. This partitioning gives good results at second order but exhibits an oscillatory behavior at higher orders. Fig. 3 shows the error in the correlation energy from order two to order twenty. As expected, HBG gives at any order a positive error. The Bloch series seems to

Table 2 Correlation energy for the ground state of N, in a minimal direct diagonalization of the full CI matrix on this basis R

2.0 2.2 2.4 2.6 2.8 3.0

HB

basis. R is the internuclear

HBG,

3rd order

4th order

3rd order

-0.1126 - 0.1191 - 0.0819 - 0.0609 - 0.0488 - 0.0370

- 0.1145 - 0.1289 - 0.0966 - 0.0723 - 0.0573 -0.0440

-0.1158 - 0.1316 -0.1017 - 0.0766 - 0.0607 - 0.0474

distance

and the exact results are given by the

Exact

-0.1168 -0.1440 - 0.1486 -0.1349 -0.1140 - 0.0904

14

F. X. Gudea, J.-L. Heully / Calculation of the Bloch effertiue Hamiltonian

Fig. 3. Convergence of the perturbation series of the ground state of N, in a minimal basis. The Bloch H* is given by the thin line, the other line giving the HB”. An Epstein-Nespet partitioning is used.

Fig. 4. Convergence of the perturbation series of the ground state of N,. The full line represent H* and HBG when the ground state only is in the model space and a Moller-Plesset partitioning is used. The broken lines give the same results but with two states in the model space.

have a periodic behavior in relative maxima. This pattern appears clearly in the HB” series. However the relative minima of HB” are closer and closer to the exact result indicating that both series must be converging. Fig. 4 shows the same calculations as before but with a Moller-Plesset partitioning [12]. The convergence is monotonous for both Hamiltonians, HBG giving better results. In order to see what happened in case of divergence we add in the model space a second state which is not coupled to the ground state but which diverges. In principle the second state should not influence the first one but since both roots are obtained from the same secular equation, the convergency of the ground state will be destroyed by the divergency of the second state. With Ha perturbation occurs already at third order and really appears at ninth order. The H “’ is much more stable, the first perturbation appears at sixth order and then it diverges rapidly by upper values at the ninth order. Size consis~~cy. The size consistency property is of primary importance for many-body systems. A size-consistent calculation is defined as the one which yields for a composite system of N non-interacting subsystems an energy which is the simple sum of the energies of the individual subsystems. In order to test the EH we used the same case as Sheppard [13].

F. X. Gadea, J.-L. Heully / Calculation of the Bloch effective Hamiltonian Table 3 Ground state energies (and error in parentheses) versus the number of subsystems (N) (second-order results). Exact answer from complete CI: 0.910156

NN

HB HBG

ss

HE HBG

cc

HE HBG

The model space is composed

H=

15

and for the two effective Hamiltonians

N=l

N+2

N=3

0.884649 (0.025507) 0.917199 (0.007043)

0.884649 (0.025507) 0.920513 (0.010357)

0.884649 (0.25507) 0.923572 (0.013416)

0.902776 (0.007380) 0.911178 (0.001022)

0.901572 (0.008584) 0.912856 (0.002700)

0.900294 (0.009862) 0.914532 (0.004376)

0.902776 (0.007380) 0.911178 (0.001022)

0.902776 (0.007380) 0.911323 (0.001167)

0.902776 (0.007380) 0.911467 (0.001311)

of N identical

subsystems

Fh(i).

i=l The Hilbert space for each subsystem is four dimensional and is spanned by the basis (n,&(n), i = 1 to 4). The Hilbert space for the composite system is the direct product of the subspaces with dimension 4N. The real matrix representation of the subsystem Hamiltonian is (cf. ref. [13]):

0.8

0.5

20.0

Calculations for N = 1, 2, 3 were performed using three successive definition of the model space: (i) The nondegenerate (NN) space only contains the zero-order ground state. (ii) The single excitation space (SS), an incomplete model space, is spanned by the N + 1 product functions which have no more than one excited subsystem &. (iii) The complete model space (CC) spanned by all product functions which have the subsystems in either & or &. The results are summarized in table 3 where the quantity E( N)/N should be constant in order to yield size consistency. An analysis of the results reveals that HBG gives always better results but that the agrees with the size-consistency conditions are never verified. The fact that H BG is not size consistent statement by Kutzelnigg [14] that size consistency is incompatible with the variational principle. However, some comments should be made on size consistency. If the error per subsystem in a non-size-consistent theory (it increases with the number of subsystems) is smaller than the error of a size-consistent method which is constant with respect to the number of subsystems), there is a value of N for which these errors are comparable. For instance, in the case presented above, supposing that the error increases linearly with the number of subsystems (which can be roughly verified in table 3) HBG and HB would become comparable for N = 6 (NN case) or for N = 45 (CC case of table 3).

16

F.X. Gadea, J.-L. Heully / Calculation of the Bloch effective Humiltonian

4. Conclusion

The traditional RS perturbation theory is based on the calculation of the wave operator. This operator is used to construct an effective Hamiltonian usually called the Bloch Hamiltonian. We have shown that with the same operator another effective Hamiltonian could be derived. Since it is an expectation value, this effective Hamiltonian presents the following advantages: (i) At the price of a little more computational effort (less than one order of perturbation) significant improvement is obtained. (ii) This Hamiltonian seems to give upper bounds for all states of the model space. This is obvious when there is only one state in the model space. In other cases we could not give the formal demonstration but all calculations provide upper bounds. Let us emphasize that the amount of computational time required to get HRG corresponds only to one multiplication of the wave operator matrix by the whole matrix of the Hamiltonian, while the determination of the wave operator at a higher order needs several similar multiplications. In connection with practical applications, the problem is that the wave operator is not usually fully determined. Especially when the second-order energy is needed, only the blocks of D which really contribute to the Bloch effective Hamiltonian are generated. In that case, the calculation of HH”‘z’ cannot be considered as efficient since the increase of calculations (determination of the whole D operator and a multiplication by H) seems not worthwhile compared with the improvement of the results (see figs. 2. 3, 4). But at higher orders. our method is expected to become competitive. For exemple HBC’ at order 3 usually gives better results than HR at order 4 (see figs. 2, 3, 4 and tables 1, 2) and is less time consuming. However, for all those advantages there is a price: the lack of size consistency.

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Berlin,

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